Abstract
We study the strong convergence of two kinds of viscosity iteration processes for approximating common fixed points of the pseudocontractive semigroup in uniformly convex Banach spaces with uniformly Gâteaux differential norms. As special cases, we get the strong convergence of the implicit viscosity iteration process for approximating common fixed points of the nonexpansive semigroup in Banach spaces satisfying some conditions. The results presented in this paper extend and generalize some results concerned with the nonexpansive semigroup in (Chen and He, 2007) and the pseudocontractive mapping in (Zegeye et al., 2007) to the pseudocontractive semigroup in Banach spaces under different conditions.
Highlights
Let E be a real Banach space with the dual space E∗ and J : E → 2E∗ be a normalized duality mapping defined byJx x∗ ∈ E∗ : x, x∗x 2 x∗ 2, 1.1 where ·, · denotes the generalized duality pairing
It is easy to see that the pseudocontractive mapping is more general than the nonexpansive mapping
Some further study concerned with approximating common fixed points of the semigroup of pseudocontractive mappings in Banach spaces, we refer to Li and Huang 16
Summary
Let E be a real Banach space with the dual space E∗ and J : E → 2E∗ be a normalized duality mapping defined byJx x∗ ∈ E∗ : x, x∗x 2 x∗ 2 , 1.1 where ·, · denotes the generalized duality pairing. Some further study concerned with approximating common fixed points of the semigroup of pseudocontractive mappings in Banach spaces, we refer to Li and Huang 16 .
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